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From Constraints to Finite Automata to Filtering Algorithms. Mats Carlsson, SICS Nicolas Beldiceanu, EMN [email protected] [email protected]. Outline. Constraint Propagation: Example & Model Constraints and Key Notions Case Study: X lex Y Definition and signature Finite automaton - PowerPoint PPT Presentation
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From Constraintsto Finite Automata
to Filtering Algorithms
Mats Carlsson, SICSNicolas Beldiceanu, EMN
ESOP, March 29, 2004
Outline Constraint Propagation: Example &
Model Constraints and Key Notions Case Study: X lex Y
Definition and signature Finite automaton Filtering algorithm
Case Study: lex_chain(X0,…,Xm-1) Definition and signature Finite automaton Filtering algorithm
Conclusion
ESOP, March 29, 2004
Example
x + y = 9 2x + 4y = 24
x 0 1 2 3 4 5 6 7 8 9
y 0 1 2 3 4 5 6 7 8 9
ESOP, March 29, 2004
Example
x + y = 9 2x + 4y = 24
x 0 1 2 3 4 5 6 7 8 9
y 0 1 2 3 4 5 6 7 8 9
ESOP, March 29, 2004
Example
x + y = 9 2x + 4y = 24
x 0 1 2 3 4 5 6 7 8 9
y 0 1 2 3 4 5 6 7 8 9
ESOP, March 29, 2004
Example
x + y = 9 2x + 4y = 24
x 0 1 2 3 4 5 6 7 8 9
y 0 1 2 3 4 5 6 7 8 9
ESOP, March 29, 2004
Example
x + y = 9 2x + 4y = 24
x 0 1 2 3 4 5 6 7 8 9
y 0 1 2 3 4 5 6 7 8 9
ESOP, March 29, 2004
Example
x + y = 9 2x + 4y = 24
x 0 1 2 3 4 5 6 7 8 9
y 0 1 2 3 4 5 6 7 8 9
ESOP, March 29, 2004
Example
x + y = 9 2x + 4y = 24
x 0 1 2 3 4 5 6 7 8 9
y 0 1 2 3 4 5 6 7 8 9
ESOP, March 29, 2004
Constraint Propagation Variables
feature variable domain (finite set of integers)
Propagators implement constraints
Propagation loop execute propagators until simultaneous
fixpoint
ESOP, March 29, 2004
Propagator Propagator p is a procedure
(coroutine) implements constraint con(p)
its semantics (set of tuples) computes on set of variables var(p)
Execution of propagator p filters domains of variables in var(p) signals failure signals entailment
ESOP, March 29, 2004
Propagators Are Intensional Propagators implement filtering
aka: narrowing, domain reduction, value removal
No extensional representation of con(p) impractical in most cases (space)
Extensional representation of constraint can be provided by special propagator often: “element” constraint, “relation” constraint,
…
ESOP, March 29, 2004
Propagation Events
Normally, a propagator p is resumed whenever some value in a domain of var(p) has been removed.
In some cases, some events (e.g. removing internal values) are irrelevant whilst other (bounds adjustments) are relevant.
ESOP, March 29, 2004
Idempotent Propagators A propagator is idempotent if it
always computes a fixpoint.
Most constraint programming systems can accommodate both idempotent and non-idempotent propagators.
ESOP, March 29, 2004
Implementing Propagators Implementation uses operations on
variables reading domain information filtering domains (removing values)
Variables are the only communication channels between propagators
Algorithms for Domain filtering Failure detection Entailment detection
ESOP, March 29, 2004
Outline Constraint Propagation: Example &
Model Constraints and Key Notions Case Study: X lex Y
Definition and signature Finite automaton Filtering algorithm
Case Study: lex_chain(X0,…,Xm-1) Definition and signature Finite automaton Filtering algorithm
Conclusion
ESOP, March 29, 2004
Classes of Constraints Basic constraints
Constraints for which the solver is complete x D, x = v, x = y (variable aliasing)
Primitive constraints (need propagators) Non-decomposable constraints
x<y, xy, x+y = z, x*y = z, …
Global constraints (need propagators) Subsume a set of basic or primitive
constraints, usually providing stronger consistency
ESOP, March 29, 2004
Support and Consistency Given: constraint C, variable x var(C),
its domain D(x), integer v. x=v has support for C iff
v D(x) C has a solution such that x=v
C is hyperarc consistent iff x var(C) v D(x) x=v has support for C
Maintaining hyperarc consistency may not be possible with polynomial algorithms (e.g. diophantine equations)
ESOP, March 29, 2004
Entailment A constraint con(p) is entailed if it
holds for any combination of values in the current domains.
Consequences for its propagator p: It has no more work to do It should not be resumed any more (up to
backtracking) It is usually reponsible for detecting
entailment
ESOP, March 29, 2004
Failure A constraint con(p) is false if it does
not hold for any combination of values in the current domains.
Consequences for its propagator p: It should signal inconsistency, e.g. by
instigating backtracking It is reponsible for detecting failure
ESOP, March 29, 2004
Notation
Vectors and subvectors X = (x0,…,xn-1) X[0,r) = (x0,…,xr-1), r n
Domain variables D(x), the domain of x (set of integers) min(x), lower bound of x, O(1) max(x), upper bound of x, O(1) prev(x,b) = max{y D(x) | y<b}, O(d) next(x,b) = min{y D(x) | y>b}, O(d)
ESOP, March 29, 2004
Constraint Signatures The constraint store is the set of all
domains D(x) For alphabet A, constraint C,
constraint store G, let S(C,G,A) be the signature of C wrt. G and A.
The filtering algorithm is derived from a finite automaton for signatures.
ESOP, March 29, 2004
Outline Constraint Propagation: Example &
Model Constraints and Key Notions Case Study: X lex Y
Definition and signature Finite automaton Filtering algorithm
Case Study: lex_chain(X0,…,Xm-1) Definition and signature Finite automaton Filtering algorithm
Conclusion
ESOP, March 29, 2004
Definition:X lex Y Let:
X = (x0,…,xn-1) Y = (y0,…,yn-1) xi, yi domain variables or integers
X lex Y holds iff n=0, or x0<y0, or x0=y0 (x1,…,xn-1) lex(y1,…,yn-1).
ESOP, March 29, 2004
Signature: X lex Y
Letter
Condition
< max(xi)<min(yi)
= xi = yi, integers
> min(xi)>max(yi)
max(xi)=min(yi)min(xi)<max(yi)
min(xi)=max(yi)max(xi)>min(yi)
? otherwise
$ End of string
ESOP, March 29, 2004
Signature example: X lex Y
X 3..3
3..4
4..5
4..5
3..3
4..4
Y 3..3
4..5
3..4
4..5
4..4
3..3
S = ? < > $
ESOP, March 29, 2004
Poset of signature letters
< = >
?
E.g., a becomes a < or a = in a ground store.
ESOP, March 29, 2004
Outline Constraint Propagation: Example &
Model Constraints and Key Notions Case Study: X lex Y
Definition and signature Finite automaton Filtering algorithm
Case Study: lex_chain(X0,…,Xm-1) Definition and signature Finite automaton Filtering algorithm
Conclusion
ESOP, March 29, 2004
Finite Automaton for X lex Y
1 2 3 4
T1 T3 T2
F1 D1 D3 D2
$
$ $ $
ESOP, March 29, 2004
Success State T1
1
T1
$
Enforce xi=yi in the leading prefix for C to hold.Afterwards, the leading prefix is ground and equal.
ESOP, March 29, 2004
Success State T2
1 2 4
T2
<q
Enforce xq<yq in order for there to be at least one < preceding the first >.
ESOP, March 29, 2004
Success State T3
1 2 3
T3
$ $
q
Only enforce xqyq , for < can appear in a later position.
ESOP, March 29, 2004
Delay States
1 2 3 4
T3 T2
D1 D3 D2
$
$ $
q
Not yet enough information to know what to do at position q.
T1
$
ESOP, March 29, 2004
Outline Constraint Propagation: Example &
Model Constraints and Key Notions Case Study: X lex Y
Definition and signature Finite automaton Filtering algorithm
Case Study: lex_chain(X0,…,Xm-1) Definition and signature Finite automaton Filtering algorithm
Conclusion
ESOP, March 29, 2004
Filtering Algorithms Non-incremental, O(n)
Run finite automaton from scratch Consider all letters from scratch
Incremental, amortized O(1) Deal with one letter change at a time Needs to know what letter has
changed, in what state
ESOP, March 29, 2004
Incremental Restart 1
1 2 3 4
T1 T3 T2
F1 D1 D3 D2
$
$ $ $
Resume in state 1.
ESOP, March 29, 2004
Incremental Restart 2
1 2 3 4
T1 T3 T2
F1 D1 D3 D2
$
$ $ $
Resume in state 2.
ESOP, March 29, 2004
Incremental Restart 3
1 2 3 4
T1 T3 T2
F1 D1 D3 D2
$
$ $ $
Resume in state 3 or 4, resp.
ESOP, March 29, 2004
Incremental Restart 4
1 2 3 4
T1 T3 T2
F1 D1 D3 D2
$
$ $ $
If changed to =, no-op. Otherwise, resume in state 3 or 4, resp.
ESOP, March 29, 2004
Finite Automaton for X <lex Y
1 2 3 4
T1 T3 T2
F1 D1 D3 D2
$ $ $ $
ESOP, March 29, 2004
Outline Constraint Propagation: Example &
Model Constraints and Key Notions Case Study: X lex Y
Definition and signature Finite automaton Filtering algorithm
Case Study: lex_chain(X0,…,Xm-1) Definition and signature Finite automaton Filtering algorithm
Conclusion
ESOP, March 29, 2004
Definition:lex_chain(X0,…,Xm-1)
Let: Xi = (xi0,…,xin-1) xij domain variables or integers
lex_chain(X0,…,Xm-1) holds iff
X0 lex … lex Xm-1
ESOP, March 29, 2004
Internal constraint:between(A,X,B) Preconditions:
A = (a0,…,an-1), B = (b0,…,bn-1) X = (x0,…,xn-1) ai,bi integers; xi domain variables i[0,n) : ai D(xi), bi D(xi)
Holds iff:A lex X lex B
ESOP, March 29, 2004
Signature:between(A,X,B)
Letter
Condition
< ai<bi next(xi,ai)bi
« ai<bi next(xi,ai)<bi
= min(xi)=ai=bi=max(xi)
# min(xi)ai=bi max(xi)ai=bi
> ai>bi bimin(xi)max(xi)ai
» ai>bi (min(xi)<biai<max(xi))
$ End of string
ESOP, March 29, 2004
Signature example: between(A,X,B)
A 5 4 6 6
X 4..6 4..6 4..6 3..7
B 5 5 4 4
S # < > » $
X’ 5..5 4..5 {4,6}
{3,4,6,7}
ESOP, March 29, 2004
Outline Constraint Propagation: Example &
Model Constraints and Key Notions Case Study: X lex Y
Definition and signature Finite automaton Filtering algorithm
Case Study: lex_chain(X0,…,Xm-1) Definition and signature Finite automaton Filtering algorithm
Conclusion
ESOP, March 29, 2004
Finite Automaton:between(A,X,B)
1 2
T1 T2
F1
=#
>»
«$ <«»#$
>=
<
State 1 denotes a prefix in which ai=bi. Hence we must enforce xi=ai=bi there.
ESOP, March 29, 2004
Success State T1:between(A,X,B)
1
T1
=#
«$Either q=n or xq=v has support for all aqvbq.Hence we enforce aqxqbq.
q
ESOP, March 29, 2004
Success State T2:between(A,X,B)
1 2
T2
=#
<«»#$
>=
<
We have:X[0,r)=A[0,r)X[0,r)=B[0,r)
Hence we enforce:aixibi for i[0,r).
Either r=n or xr=v has support for all vbr var.
Hence we enforce:xrbr xrar.
r
ESOP, March 29, 2004
Outline Constraint Propagation: Example &
Model Constraints and Key Notions Case Study: X lex Y
Definition and signature Finite automaton Filtering algorithm
Case Study: lex_chain(X0,…,Xm-1) Definition and signature Finite automaton Filtering algorithm
Conclusion
ESOP, March 29, 2004
Feasible Upper Bound:Problem Given X and B, compute the
lexicographically largest U such that:
U lex B i[0,n) : ui D(xi)
Similarly for feasible lowest bound
ESOP, March 29, 2004
Feasible Upper Bound:Algorithm
Compute as the smallest i such thatU[0,i)B[0,i)
Compute: ui = bi, if i < ui = prev(xi,bi), if i =
ui = max(xi), if i > Similarly for feasible lowest bound
ESOP, March 29, 2004
Filtering:lex_chain(X0,…,Xm-1)
1. Compute feasible upper bound Bi for Xi from i=m-1 down to i=0.
2. Compute feasible lower bound Ai for Xi from i=0 to i=m-1.
3. Enforce Ai lex Xi lex Bi for all i.
Arc-consistency in O(nmd) time.
ESOP, March 29, 2004
Outline Constraint Propagation: Example &
Model Constraints and Key Notions Case Study: X lex Y
Definition and signature Finite automaton Filtering algorithm
Case Study: lex_chain(X0,…,Xm-1) Definition and signature Finite automaton Filtering algorithm
Conclusion
ESOP, March 29, 2004
Results An approach to designing filtering
algorithms by derivation from FAs on constraint signatures
Case studies and hyperarc consistency algorithms for two constraints: X lex Y, running in amortized O(1) time per
propagation event lex_chain(X0,…,Xm-1), running in O(nmd)
time per invocation
ESOP, March 29, 2004
Future Work What constraints are amenable to the
approach? Where does the alphabet come from? Where does the automaton come from? Where do the pruning rules come from? How do we make the algorithms
incremental?
ESOP, March 29, 2004
References and proofs SICS T2002-17: Revisiting the
Lexicographic Ordering Constraint, Mats Carlsson, Nicolas Beldiceanu.
SICS T2002-18: Arc-Consistency for a Chain of Lexicographic Ordering Constraints, Mats Carlsson, Nicolas Beldiceanu.
http://www.sics.se/libindex.html
ESOP, March 29, 2004
Related Work Global constraints for
lexicographic orderings. A. Frisch, B. Hnich, Z. Kızıltan, I. Miguel, T. Walsh. Proc. CP’2002. LNCS 2470, pp. 93-108, Springer, 2002.